Problem 26 - College Algebra

Question

Find the standard form of the equation of each parabola satisfying the given conditions. Vertex: \((5,-2) ;\) Focus: \((7,-2)\)

Step-by-Step Solution

Answer

The standard form of the parabolic equation is \((y+2)^2 = 8(x-5)\)

Step 1: Identify the shift

From the problem, the vertex \((h = 5, k = -2)\) will shift the graph horizontally and vertically.

Step 2: Identify the direction and compute a

The Focus \((7,-2)\) lies to the right of the vertex which indicates the parabola needs to open to the right. Knowing the vertex and the focus, one can find a as the difference in the x-coordinates of these two points. Thus, a = 7 - 5 = 2.

Step 3: Write the equation

Using the standard form equation of a horizontal parabola (y - k)² = 4a(x - h), substitute h = 5, k = -2, a = 2 to get the equation of the parabola: \((y + 2)^2 = 4*2*(x - 5)\), which simplifies to \( (y+2)^2 = 8(x-5) \)